Theorem pnrmtop 23381

Description: A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmtop	⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Proof of Theorem pnrmtop

Step	Hyp	Ref	Expression
1		pnrmnrm 23380	. 2 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
2		nrmtop 23376	. 2 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
3	1, 2	syl 17	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2141  Topctop 22933  Nrmcnrm 23350  PNrmcpnrm 23352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-cnv 5653  df-dm 5655  df-rn 5656  df-iota 6473  df-fv 6525  df-ov 7395  df-nrm 23357  df-pnrm 23359

This theorem is referenced by:  pnrmopn  23383